Kerodon

Example 9.5.6.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be essentially small $\infty $-categories having cocompletions $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$, respectively. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{D}}}$ be its cocontinuous extension. Then $F$ admits right adjoint $G: \widehat{\operatorname{\mathcal{D}}} \rightarrow \widehat{\operatorname{\mathcal{C}}}$, which we can identify with the functor $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ given by precomposition with $f^{\operatorname{op}}$ (Example 8.4.4.5). If $f$ is a localization functor (Definition 6.3.3.1), then $G$ is fully faithful, so that $F$ is a Bousfield localization functor.